(ed.). Gravitational waves (IOP, 2001)(422s).

Application of the TT gauge to the mass quadrupole field 63• the only motions that produce the radiation are the ones transverse to the lineof sight; and• the induced motions in a detector mirror the motions of the source projectedonto the plane of the sky.To see why these are true, we define the transverse traceless quadrupole tensorM TTij =⊥ k i ⊥ l j M kl − 1 2 ⊥ ij⊥ kl M kl . (6.27)(Notice that some of our definitions of tracelessness involve subtracting 1 3of thetrace, as in equation (6.24), and sometimes 1 2of the trace, as in equation (6.27).The appropriate factor is determined by the effective dimensionality (rank) ofthe tensor. Although we have three spatial dimensions, the projection tensor ⊥projects the mass-quadrupole tensor onto a two-dimensional plane, where thetrace involves only two components, not three.)Now, if in equation (6.26) we replace ˜M ij by its definition in terms of M ij ,and then collect terms appropriately, it is not hard to show that the equationsimplifies to its most natural form:¯h TTij = 2 r ¨M TTij . (6.28)This could of course have been derived directly by applying the TT operation toequations (6.9)–(6.11). Since this equation involves only the TT part of M, ourfirst assertion above is proved. According to this equation, in order to calculatethe quadrupole radiation that a particular observer will receive, one need onlycompute the mass-quadrupole tensor’s second time-derivative, project it onto theplane of the sky as seen by the observer looking toward the source, take away itstrace, and rescale it by a factor 2/r. In particular, the TT tensor that describes theaction of the wave (as in the polarization diagram in figure 2.1) is a copy of theTT tensor of the mass distribution. This proves our second assertion above.Looking again at figure 2.1 we imagine a detector consisting of two freemasses whose separation is being monitored. If the wave causes them to oscillaterelative to one another along the x-axis (the ⊕ polarization), this means that thesource motion contained a component that did the same thing. If the source is abinary, then the binary orbit projected onto the sky must involve motion of thestars back and forth along either the x- or the y-axis.It is possible from this to understand many aspects of quadrupole radiationin a simple way. Consider a binary star system with a circular orbit. Seen by adistant observer in the orbital plane, the projected source motion is linear, backand forth. The received polarization will be linear, the polarization ellipse alignedwith the orbit. Seen by a distant observer along the axis of the orbit of thebinary, the projected motion is circular, which is a superposition of two linearmotions separated in phase by 90 ◦ . The received radiation will also have circularpolarization. Because both linear polarizations are present, the amplitude of the